[Paper Review] Steady-state mode III cracks in a viscoelastic lattice model

Leonid Pechenik, Herbert Levine+1 more|arXiv (Cornell University)|Feb 20, 2000

Ultrasonics and Acoustic Wave Propagation25 references3 citations

TL;DR

This paper extends Slepyan's lattice model for steady-state mode III crack propagation to include Kelvin viscosity, using the Wiener-Hopf method to analyze viscoelastic effects. It identifies a critical crack velocity beyond which the steady-state solution becomes inconsistent due to additional bond-breaking, demonstrating that dissipation delays but does not eliminate microbranching instabilities—key for understanding dynamic fracture beyond continuum elasticity.

ABSTRACT

We extend the Slepyan solution of the problem of a steady-state crack in an infinite ideally brittle lattice model to include dissipation in the form of Kelvin viscosity. As a demonstration of this technique, based on the Wiener-Hopf method, we apply the method to mode III cracks in a square lattice. We use this solution to find the critical velocity at which the steady-state solution becomes inconsistent due to additional bond-breaking; this point signaling the onset of complex dynamical behavior.

Motivation & Objective

• To extend the idealized brittle lattice model of crack propagation to include dissipative effects via Kelvin viscosity.
• To analyze how viscosity affects the stability of steady-state crack motion in a square lattice.
• To determine the critical crack velocity at which the traveling wave solution becomes inconsistent due to additional bond-breaking.
• To demonstrate that dissipation delays, but does not eliminate, the onset of complex dynamical behavior such as microbranching.
• To provide a framework applicable to more realistic lattice geometries, such as triangular lattices, in future work.

Proposed method

• Uses a square lattice model with mass points connected by linear springs that break irreversibly at elongation 2ε.
• Introduces Kelvin viscosity η to each spring, modeling energy dissipation proportional to spring strain rate.
• Applies the Wiener-Hopf technique to solve the infinite lattice problem for steady-state crack motion.
• Imposes a local driving force on the crack surface to replace external boundary conditions, enabling analytical treatment.
• Derives the velocity-driving curve from the formal solution and evaluates it numerically.
• Analyzes self-consistency of the traveling wave solution by checking whether spring displacements exceed the breaking threshold.

Experimental results

Research questions

• RQ1 How does the inclusion of Kelvin viscosity affect the steady-state crack velocity in a brittle lattice model?
• RQ2 At what critical velocity does the steady-state solution become inconsistent due to additional bond-breaking?
• RQ3 Can dissipation delay the onset of microbranching instabilities observed in dynamic fracture?
• RQ4 How does the strength of the bonds along the crack path (parameterized by k) influence the critical velocity?
• RQ5 To what extent does the viscous damping alter the applicability of the Yoffe criterion to mode III cracks?

Key findings

• The critical velocity at which the steady-state solution becomes inconsistent increases with increasing viscosity η, indicating a delaying effect of dissipation on instability onset.
• For k=1 (standard bonds), the critical velocity v_cr increases with η, but the instability still occurs at finite η, meaning dissipation does not eliminate the instability.
• When bond strength is reduced (k<1), the critical velocity increases further, suggesting that weakened bonds can enable stable crack propagation at higher speeds.
• The onset of inconsistency is strongly dependent on η, while the underlying continuum elastic field remains independent of viscosity.
• Numerical results in Figures 5 and 6 show that both v_cr and τ_cr (critical time) increase with η for k=1, 0.75, and 0.5, confirming the stabilizing role of viscosity.
• The results imply that supersonic or intersonic crack speeds may be possible in finite-width systems with sufficiently weakened bonds, especially under viscous damping.